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Pythagoras0 (토론 | 기여) (→메타데이터) |
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| − | + | ==개요== | |
| − | + | * [[스토크스 정리]]의 특수한 경우:<math>\iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, {d}A=\oint_{\partial D} (P\, {d}x + Q\, {d}y)</math> | |
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| − | < | + | ==폐곡선에 둘러싸인 영역의 넓이== |
| + | * 폐곡선 C에 둘러싸인 영역의 넓이는 다음 공식으로 주어진다 :<math>A=\oint_{C} x dy = \oint_{C} - y dx =\frac{1}{2}\oint_{C} x dy-y dx</math> | ||
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| − | + | ===증명=== | |
| + | 면적은 <math>A= \iint_{D} 1 \, {d}A</math>으로 주어지므로, 그린 정리를 이용하여 다음 각각의 경우 <math>P,Q</math> 가 <math>\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)=1</math>을 만족함을 보이면 된다. | ||
| − | + | *<math>P=0,Q=x</math> | |
| + | *<math>P=-y,Q=0</math> | ||
| + | *<math>P=-y/2,Q=x/2</math> | ||
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| − | * | + | ==꼭지점이 주어진 다각형의 넓이== |
| − | + | * 평면위의 점 <math>P_i=(x_i,y_i), i=0,1,\cdots, n-1</math>을 꼭지점으로 갖는 n-각형 <math>\overline{P_0P_1\cdots P_{n-1}}</math>의 넓이 <math>A</math>는 다음으로 주어진다 :<math>A=\frac{1}{2}\sum_{i=0}^{n-1}x_iy_{i+1}-y_ix_{i+1}</math> 이 때, <math>(x_{n},y_{n})=(x_{0},y_{0}).</math> 이다 | |
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| − | + | ==역사== | |
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* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | * [[ | + | * [[수학사 연표]] |
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| − | + | ==메모== | |
| − | + | * [http://www.youtube.com/watch?v=pvGuGaImTek Digital planimeter demonstration ] | |
| + | * [http://www.mathematik.com/Planimeter/explanation.html How does the planimeter work?] | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxZThyY3Rtbk9BMFE/edit?usp=drivesdk | ||
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| − | + | ==관련된 항목들== | |
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| − | + | * [[각원소 벡터장|각원소벡터장]] | |
| + | * [[선적분]] | ||
| + | * [[타원의 넓이]] | ||
| + | * [[픽의 정리(Pick's Theorem)]] | ||
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| − | + | ==사전 형태의 자료== | |
| − | * | + | * http://ko.wikipedia.org/wiki/그린정리 |
* http://en.wikipedia.org/wiki/Green_theorem | * http://en.wikipedia.org/wiki/Green_theorem | ||
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| − | + | ==관련논문== | |
| − | * [http://www.jstor.org/stable/2689760 Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years] | + | * [http://www.jstor.org/stable/2689760 Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years] |
** Thomas Archibald, , Math. Mag. 62 (1989), 219-232 | ** Thomas Archibald, , Math. Mag. 62 (1989), 219-232 | ||
* http://www.jstor.org/action/doBasicSearch?Query= | * http://www.jstor.org/action/doBasicSearch?Query= | ||
* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet | ||
* http://dx.doi.org/ | * http://dx.doi.org/ | ||
| + | [[분류:미적분학]] | ||
| − | + | == 노트 == | |
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| − | + | ===말뭉치=== | |
| + | # Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).<ref name="ref_bb3b58b6">[https://mathinsight.org/greens_theorem_idea#:~:text=Green's%20theorem%20says%20that%20if,the%20macroscopic%20circulation%20around%20C. The idea behind Green's theorem]</ref> | ||
| + | # Green's theorem and other fundamental theorems Green's theorem is one of the four fundamental theorems of vector calculus all of which are closely linked.<ref name="ref_bb3b58b6" /> | ||
| + | # Are you ready to use Green's theorem?<ref name="ref_bb3b58b6" /> | ||
| + | # Make sure you understand when you are allowed to use Green's theorem, check out some other ways of writing Green's theorem, then investigate some examples.<ref name="ref_bb3b58b6" /> | ||
| + | # Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses.<ref name="ref_512e2c21">[https://brilliant.org/wiki/greens-theorem/ Brilliant Math & Science Wiki]</ref> | ||
| + | # Note that Green's Theorem applies to regions in the xy-plane.<ref name="ref_bdba4266">[http://www.math.lsa.umich.edu/~glarose/classes/calcIII/web/17_4/ green's theorem]</ref> | ||
| + | # We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.<ref name="ref_bdba4266" /> | ||
| + | # The term Green's theorem is applied to a collection of results that are really just restatements of the fundamental theorem of calculus in higher dimensional problems.<ref name="ref_728ef97a">[https://wiki.seg.org/wiki/Green%27s_theorem Green's theorem]</ref> | ||
| + | # The various forms of Green's theorem includes the Divergence Theorem which is called by physicists Gauss's Law, or the Gauss-Ostrogradski law.<ref name="ref_728ef97a" /> | ||
| + | # We can also write Green's Theorem in vector form.<ref name="ref_07694272">[https://math24.net/greens-theorem.html Green’s Theorem]</ref> | ||
| + | # Subsection 12.7.2 Green's Theorem So far in this section, we have restricted ourselves to relatively nice closed curves when thinking about circulation.<ref name="ref_a71c3b35">[https://activecalculus.org/vector/S_Vector_GreensTheorem.html Green's Theorem]</ref> | ||
| + | # The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.<ref name="ref_a71c3b35" /> | ||
| + | # } \end{equation*} In Activity 12.7.2, we showed that Green's Theorem holds when the region \(R\) is a rectangle with sides parallel to the coordinate axes.<ref name="ref_a71c3b35" /> | ||
| + | # The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.<ref name="ref_a71c3b35" /> | ||
| + | ===소스=== | ||
| + | <references /> | ||
| − | + | == 메타데이터 == | |
| − | + | ===위키데이터=== | |
| − | + | * ID : [https://www.wikidata.org/wiki/Q321237 Q321237] | |
| − | * | + | ===Spacy 패턴 목록=== |
| − | + | * [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}] | |
| − | * [ | ||
2022년 8월 11일 (목) 21:34 기준 최신판
개요
- 스토크스 정리의 특수한 경우\[\iint_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\, {d}A=\oint_{\partial D} (P\, {d}x + Q\, {d}y)\]
폐곡선에 둘러싸인 영역의 넓이
- 폐곡선 C에 둘러싸인 영역의 넓이는 다음 공식으로 주어진다 \[A=\oint_{C} x dy = \oint_{C} - y dx =\frac{1}{2}\oint_{C} x dy-y dx\]
증명
면적은 \(A= \iint_{D} 1 \, {d}A\)으로 주어지므로, 그린 정리를 이용하여 다음 각각의 경우 \(P,Q\) 가 \(\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)=1\)을 만족함을 보이면 된다.
- \(P=0,Q=x\)
- \(P=-y,Q=0\)
- \(P=-y/2,Q=x/2\)
꼭지점이 주어진 다각형의 넓이
- 평면위의 점 \(P_i=(x_i,y_i), i=0,1,\cdots, n-1\)을 꼭지점으로 갖는 n-각형 \(\overline{P_0P_1\cdots P_{n-1}}\)의 넓이 \(A\)는 다음으로 주어진다 \[A=\frac{1}{2}\sum_{i=0}^{n-1}x_iy_{i+1}-y_ix_{i+1}\] 이 때, \((x_{n},y_{n})=(x_{0},y_{0}).\) 이다
역사
메모
매스매티카 파일 및 계산 리소스
관련된 항목들
사전 형태의 자료
관련논문
- Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years
- Thomas Archibald, , Math. Mag. 62 (1989), 219-232
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/
노트
말뭉치
- Hence, Green's theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above).[1]
- Green's theorem and other fundamental theorems Green's theorem is one of the four fundamental theorems of vector calculus all of which are closely linked.[1]
- Are you ready to use Green's theorem?[1]
- Make sure you understand when you are allowed to use Green's theorem, check out some other ways of writing Green's theorem, then investigate some examples.[1]
- Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses.[2]
- Note that Green's Theorem applies to regions in the xy-plane.[3]
- We could evaluate the line integral of F.dr along C directly, but it is almost always easier to use Green's theorem.[3]
- The term Green's theorem is applied to a collection of results that are really just restatements of the fundamental theorem of calculus in higher dimensional problems.[4]
- The various forms of Green's theorem includes the Divergence Theorem which is called by physicists Gauss's Law, or the Gauss-Ostrogradski law.[4]
- We can also write Green's Theorem in vector form.[5]
- Subsection 12.7.2 Green's Theorem So far in this section, we have restricted ourselves to relatively nice closed curves when thinking about circulation.[6]
- The restriction that the curve in Green's Theorem prohibits curves such as the one below, which crosses itself.[6]
- } \end{equation*} In Activity 12.7.2, we showed that Green's Theorem holds when the region \(R\) is a rectangle with sides parallel to the coordinate axes.[6]
- The discussion that introduced the previous subsection may have you convinced that Green's Theorem holds in its full form.[6]
소스
메타데이터
위키데이터
- ID : Q321237
Spacy 패턴 목록
- [{'LOWER': 'green'}, {'LOWER': "'s"}, {'LOWER': 'theorem'}]